On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

Djalil Chafai (1) , Joseph Lehec (2, 1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris

This note, mostly expository, is devoted to Poincaré and logarithmic Sobolev inequalities for a class of singular Boltzmann-Gibbs measures. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from the convexity of confinement and interaction. We prove optimality in the case of quadratic confinement by using a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gauss-ian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics which admits the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the McKean-Vlasov mean-field limit of the dynamics, as well as the consequence of the logarithmic Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]

Djalil Chafai, Joseph Lehec. On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures. 2018. ⟨hal-01781502v2⟩